I have matrix $A$ that is $(M \times M)$ square matrix, $x$ $(M \times N)$ matrix, $b$ is $(M \times N)$ matrix. Knowing $A$ and $b$ I would like to get the $x$ from the equation $Ax=b$. $N=p \times q$, so consider $x$ as an $M$ number of $p \times q$ pixel images. These images are sparse when we take total variational $TV(x)$. I would like to solve the following minimization problem for $x$;
Define an operator $g(x)=\parallel x^T\parallel_2,$ or $g(x)=\parallel x^T\parallel_1,$ that maps $M \times N \mapsto N \times 1$.
$$\min \frac{1}{2}\parallel Ax-b\parallel^2_2+k\parallel z\parallel_1,$$ subject to $Fg(x)-z=0$.
$F$ is the difference matrix to take numerical gradient of pixels.
Unlike in the deblurring+denoising problem, my matrix $A$ and $x$ are in different sizes.
The ADMM (Alternating Direction Method of Multipliers) solution to my minimization problem is given as:
$$x^{k+1}=(A^TA+pF^TF)^{-1}(A^Tb+pF^T(z^k-u^k))$$
$$z^{k+1}=S_{t/p}(Fx^{k+1}+u^{k})$$
$$u^{k+1}=u^k+Fx^{k+1}-z^{k+1}$$
When calculating $x^{k+1}$ the matrix dimensions are not compatible in my case. Gradient matris should be applied to total number of pixels $N$, $F$ matrix should have $N$ columns. So the term $(A^TA+pF^TF)$ is not proper way to add 2 matrices.
How can I overcome this and find a solution to my problem?